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Consistency of high-dimensional AIC-type and Cp-type criteria in multivariate linear regression

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  • Fujikoshi, Yasunori
  • Sakurai, Tetsuro
  • Yanagihara, Hirokazu

Abstract

The AIC, the multivariate Cp and their modifications have been proposed for multivariate linear regression models under a large-sample framework when the sample size n is large, but the dimension p of the response variables is fixed. In this paper, first we propose a high-dimensional AIC (denoted by HAIC) which is an asymptotic unbiased estimator of the risk function defined by the expected log-predictive likelihood or equivalently the Kullback–Leibler information under a high-dimensional framework p/n→c∈[0,1). It is noted that our new criterion provides better approximations to the risk function in a wide range of p and n. Recently Yanagihara et al. (2012) [17] noted that AIC has a consistency property under Ω=O(np) when p/n→c∈[0,1), where Ω is a noncentrality matrix. In this paper we show that several criteria including HAIC and Cp have also a consistency property under Ω=O(n) as well as Ω=O(np) when p/n→c∈[0,1). Our results are checked numerically by conducting a Monte Carlo simulation.

Suggested Citation

  • Fujikoshi, Yasunori & Sakurai, Tetsuro & Yanagihara, Hirokazu, 2014. "Consistency of high-dimensional AIC-type and Cp-type criteria in multivariate linear regression," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 184-200.
    • Handle: RePEc:eee:jmvana:v:123:y:2014:i:c:p:184-200
    DOI: 10.1016/j.jmva.2013.09.006
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    References listed on IDEAS

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    1. Yanagihara, Hirokazu & Satoh, Kenichi, 2010. "An unbiased Cp criterion for multivariate ridge regression," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1226-1238, May.
    2. James R. Schott, 2005. "Testing for complete independence in high dimensions," Biometrika, Biometrika Trust, vol. 92(4), pages 951-956, December.
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    Cited by:

    1. Fujikoshi, Yasunori & Sakurai, Tetsuro, 2016. "High-dimensional consistency of rank estimation criteria in multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 199-212.
    2. Yasunori Fujikoshi & Tetsuro Sakurai, 2023. "High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures," Mathematics, MDPI, vol. 11(3), pages 1-15, January.
    3. Fujikoshi, Yasunori, 2022. "High-dimensional consistencies of KOO methods in multivariate regression model and discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    4. Yanagihara, Hirokazu & Oda, Ryoya & Hashiyama, Yusuke & Fujikoshi, Yasunori, 2017. "High-dimensional asymptotic behavior of the difference between the log-determinants of two Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 70-86.
    5. Imori, Shinpei & Rosen, Dietrich von, 2015. "Covariance components selection in high-dimensional growth curve model with random coefficients," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 86-94.
    6. Mori, Yuichi & Suzuki, Taiji, 2018. "Generalized ridge estimator and model selection criteria in multivariate linear regression," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 243-261.

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